Estimation of Agent-Based Models:
the case of an Asymmetric Herding Model∗
Simone Alfaranoa , Friedrich Wagnerb and Thomas Luxa
July 5, 2004
a
b
Dept. of Economics, University of Kiel, Germany
Dept. of Physics, University of Kiel, Germany
Abstract
The behavioral origins of the stylized facts of financial returns have been addressed in a growing body
of agent-based models of financial markets. While the traditional efficient market viewpoint explains all
statistical properties of returns by similar features of the news arrival process, the more recent behavioral finance models explain them as imprints of universal patterns of interaction in these markets. In
this paper we contribute to this literature by introducing a very simple agent-based model in which the
ubiquitous stylized facts (fat tails, volatility clustering) are emergent properties of the interaction among
traders. The simplicity of the model allows us to estimate the underlying parameters, since it is possible
to derive a closed form solution for the distribution of returns. We show that the tail shape characterizing
the fatness of the unconditional distribution of returns can be directly derived from some structural variables that govern the traders’ interactions, namely the herding propensity and the autonomous switching
tendency.
Keywords: Herd Behavior; Speculative Dynamics; Fat Tails; Volatility Clustering.
JEL Classification: G12; C61.
Corresponding author:
Dept. of Economics, University of Kiel; Olshausenstr. 40, D-24118 Kiel, Germany.
E-mail: [email protected]
∗
Earlier versions of this paper have been presented at the 11th Symposium of the Society of Nonlinear Dynamics
and Econometrics, Florence March 2003, the 8th Spring Meeting of Young Economists, Leuven April 2003, the 8th
Workshop on Economics with Heterogeneous Interacting Agents, Kiel May 2003, the 27th congress of Associazione
per la Matematica Applicata alle Scienze Economiche e Sociali, Cagliari September 2003; research seminars at
the department of Econometrics, University of Geneva, March 2003, and at the department of Physics, University
of Cagliari, May 2003, and have gained considerably from comments by many participants in these events. We
are particulary grateful to Manfred Gilli and Tony He for helpful suggestions and intense discussions during the
preparation of this paper.
1
Introduction
Research in empirical finance has come up with a number of stylized facts of empirical data,
involving conditional and unconditional properties of the time series. Starting with the former,
the non-Gaussian character of the unconditional distribution of returns is a well-known ubiquitous finding in the literature [26, 23, 31]. In particular, it has been observed that practically all
financial time series are leptokurtic [26], i.e. they are characterized by a higher concentration of
probability mass in the center and in the tail of the distribution than the Gaussian. However,
kurtosis is not a fully adequate measure of deviations from normality, since existence of the forth
moment itself depends on the tail behavior of the distribution. It has been recently recognized
that the extreme part of the distribution is well approximated by a Pareto law [31, 27] (in contrast to the exponential decay of the Gaussian). This indicates that a more appropriate way
to measure the dispersion of returns is the tail index, defined as the order of the highest finite
absolute moment. An analysis of estimated indices across different markets and time resolutions
[22, 31] shows a relatively small interval of variability of these estimates, which typically hover
between 2.5 and 4. Gopikrishnan et al. [13] go even further claiming that with the typical
estimates around 3 an inverse cubic law holds for financial fluctuations.
Another universal feature that is strongly related to the leptokurtic shape of the distribution
of returns is the intermittent behavior of the volatility, or volatility clustering, phenomenologically described as “periods of quiescence and turbulence tending to cluster together” [31, 35].
While serial correlation is absent in raw returns (which is in harmony with informational
efficiency of financial markets), volatility clustering leads to positive autocorrelation for absolute
and squared returns over an extended time horizon [31]. The slow decay of the auto-correlation
of all measures of volatility1 is again well represented by a power law, which is the defining
property of long memory stochastic processes. It has been shown that the hyperbolic decline of
the autocorrelation of the volatility is another extremely robust empirical phenomenon [21].
An additional important item to the list of empirical regularities is the so called unit root
property [31, 35]: one is not able to reject the hypothesis that the prices follow a random walk
or a martingale process. The resulting lack of predictability of the future price is in accordance
with informational efficiency of the market.
Despite the huge amount of empirical work on the statistical properties of financial data,
the origin of the universality of fat tails and volatility clustering in financial markets is still
obscure. One could distinguish two competing hypotheses for their origin: one derived from the
traditional Efficient Market Hypothesis (EMH) and a recent alternative which we might call the
Interacting Agent Hypothesis (IAH). The EMH states that the price fully and instantaneously
reflects any new information: the market is, therefore, efficient in aggregating available information with his invisible hand. The agents are assumed to be rational and homogeneous with
respect to the access and their assessment of information; and, as a consequence, interactions
among them can be neglected. The returns are, therefore, a mere reflection of forthcoming
information so that, consequently, their empirical regularities would simply mirror those of the
1
Volatility can be measured, for instance, by the absolute or the squared value of the returns, local variance
or standard deviation.
2
“news arrival process”, confining their origin to forces outside the realm of economics.
However, in recent literature, several papers try to explain the stylized facts as the macroscopic outcome of an ensemble of heterogeneous interacting agents [24, 6, 7]. In this view, the
market is populated by agents with different characteristics, such as, differences in access to and
interpretation of available information, different expectations, or different trading strategies.
The traders interact, for example, by exchanging information or they trade imitating the behavior of others. The market possesses, then, an endogenous dynamics, and the strict one-to-one
relationship with the news arrival process does not hold any longer (although the market might
still be efficient in the sense of a lack of predictability). The universality of the statistical regularities is seen as an emergent property of this internal dynamics, governed by the interactions
among agents. One of the main drawbacks of this approach arises from the complexity of the
models which typically prevents an analytical solution and leaves only the possibility of a rough
calibration of the key parameters (e.g. LeBaron [20]). No wonder, empirical implementations of
agent-based models are practically non-existent. While earlier models of chartist-fundamentalist
interaction with simple trading rules have been tested (with mixed success) via regime-switching
time series models (cf. [30, 34]), for true multi-agent models parameter estimation and evaluation on the base of empirical data is still largely missing in the pertinent literature. As far as we
know the only exception is a recent paper by Gilli and Winker [12], who estimate some of the
parameters of Kirman’s seminal herding model [17, 18]. They introduce a simulated moment
approach extracting two key parameters of the model via matching of the empirical kurtosis and
the first autocorrelation coefficient of squared returns. The main finding of their application to
daily DEM/$ data is that estimated parameters are such that majorities would emerge in the
herding process, instead of a balanced distribution of agents on the two groups of chartists and
fundamentalists.
The goal of this paper is to add to the direction of empirical research on agent-based models
initiated by Gilli and Winker: we set the stage by formulating an extremely simple agent-based
model which, however, is still able to reproduce the key stylized facts listed above, in line with
previous papers [32, 2]. The main contribution of the present work, then, consists in a direct
estimation of the underlying parameters of the model using a parametric approach. Our traders
are also divided into the groups of fundamentalists and noise traders following the legacy of
Beja and Goldman [4], as well as many other authors. Like in [12] the interaction among the
traders is based on a variant of the herding mechanism introduced by Kirman, but this herding
mechanism is embedded into a simpler framework for the market dynamics which allows to derive a closed-form solution for the distribution of returns. The simplicity of this approach also
enables us to consider a generalization of the herding mechanism of our model by allowing for
asymmetric transition probabilities.
As it turns out, the resulting structure of our model is analogous to that of stochastic volatility models, which were introduced as a generalization of the Brownian motion in Black and
Scholes’s option pricing theory [5]. However, although these models can reproduce several empirical regularities, they are nevertheless driven more by mathematical convenience rather than
economic intuition. We attempt, on the contrary, to introduce a plausible behavioral framework
in our model, allowing for an economic interpretation of the resulting dynamics. It turns out
that returns are governed by the product of a stochastic volatility term, related to the change
3
in the strategies among traders, and a multiplicative iid noise term linked to the noise traders’
misperception of the fundamental price.
The paper is organized as follows: in section 2 we describe the asymmetric herding mechanism
that is employed as the main ingredient in our agent-based financial market. The complete
framework of our speculative market is introduced in section 3. The estimation procedure for
the parameters is detailed in section 4, and some final remarks conclude the paper.
2
The herding mechanism
Kirman [18] has introduced a simple stochastic model of information transmission initially designed to explain the herding behavior in ant colonies, gathering food from two identical sources
located in their neighborhood. His formalization is symmetric with respect to switching probabilities generating a symmetric probability distribution of ants’ visiting times at both mangers.
In a long chain of subsequent papers this and similar approaches have been applied as models of
herding and contagion phenomena in financial markets ([17],[19],[25]). Costantini and Garibaldi
[8] have derived the equilibrium distribution of the related discrete-time stochastic process within
the more general theoretical framework of Polya urn processes. Alfarano et al. [2] have derived
and solved the associated Fokker-Planck equation for the pertinent continuous symmetric dynamic process. In the following, we generalize the previous approach by considering asymmetric
transition probabilities that could describe, for instance, scenarios with non-identical sources of
food or biased herding tendencies in a speculative market setting.
We describe the dynamics of the market as a jump Markov process in continuous time. The
market is populated by a fixed number of agents N , each of them being either in state 1 or in
state 2. The number of agents in the first state is denoted by n, and that in the second state
by N − n. The conditional probabilities ρ̄(n0 , t + ∆t|n, t) of changing from n at time t to n0 at
time t + ∆t with n0 − n = ±1, 0 by a single switch are related to the transition probabilities
π(n → n0 ) per unit time by ρ̄(n0 , t + ∆t|n, t) = ∆t π(n → n0 ). The latter are given by
π(n → n + 1) = (N − n)(a1 + bn) and π(n → n − 1) = n[a2 + b(N − n)].
(1)
The constants a1 and a2 describe the idiosyncratic propensity to change the state, while the
term b encapsulates the herding tendency. In contrast to earlier contributions, we allow the
two constant parameters, a1 and a2 , to assume different values, generating, then, the required
asymmetric behavior.
P The transition probability π(n → n) to remain in the same state follows
from the condition n0 ρ̄(n0 , t + ∆t|n, t) = 1. Note that since transition probabilities should
remain restricted to values < 1 per time increment, discrete simulations of the model would
only be possible up to an upper limit of ∆t which is given, for large N , by
∆t =
2
.
bN 2
(2)
The transition probabilities imply the so-called Master equation for the probability2 ω̄n (t) to
2
We denote probabilities referring to n by ω̄ to distinguish them from the probability densities ω(z) for the
continuous variable z = n/N . Both are related by ω(n/N ) = N ω̄n
4
find n agents in the state 1 at time t
∆ω̄n (t) X
=
ω̄n0 π(n0 → n) − ω̄n π(n → n0 ) .
∆t
0
(3)
n
For large enough N we can represent the group dynamics by a continuous variable:
n
.
N
z=
(4)
As shown in appendix A1, the Master equation (3) is equivalent to a Fokker-Planck equation if
terms of order 1/N 2 are neglected
∂ω(z, t)
∂
1 ∂2
= − [A(z)ω(z, t)] +
[D(z)ω(z, t)]
∂t
∂z
2 ∂z 2
(5)
where A(z) represents the drift term
A(z) = a1 − (a1 + a2 )z
(6)
and D(z) the diffusion term:
1
(a1 (1 − z) + a2 z).
N
D(z) = 2b(1 − z)z +
(7)
It is remarkable that A(z) is independent of the herding mechanism and that the 1/N corrections
appear only in the diffusion term.
In appendix A2 we compute the equilibrium distribution ω0 (z, N ). Interestingly, it turns
out that it depends only on the ratios
ε1 =
a1
b
and ε2 =
a2
b
but not on the size of the constants ai and b:
ε1 ε1 −1 ε2 ε2 −1
ω0 (z, N ) = K(ε1 , ε2 , N ) z +
1−z+
.
2N
2N
(8)
If we neglect the 1/N terms in D(z) we obtain
ω0 (z) =
1
z ε1 −1 (1 − z)ε2 −1
B(ε1 , ε2 )
where
B(ε1 , ε2 ) =
Γ(ε1 )Γ(ε2 )
.
Γ(ε1 + ε2 )
(9)
(10)
We see from (8) and (9) that for ε1 < 1 or ε2 < 1 the probability density ω0 (z, N ) does not
converge uniformly to ω0 (z). For z ∼ 1/N or 1 − z ∼ 1/N the 1/N term in D has to be kept
in order to obtain a finite (however large) value of the probability density.3 In the following, we
3
It is worth pointing out that, despite this N-dependence of ω0 , the qualitative dynamics of our model is
independent of N (cf. Alfarano, Wagner and Lux [2]). The importance of system size N in eq. (8) for ε1,2 < 1
stems solely from the relevance of the grid of admissible values of z in scenarios with a bi-modal distribution and
concentration of probability mass at the upper and lower boundaries of its support rather than from a “true”
qualitative dependence on system size.
5
ignore the 1/N term in D and the correspondent N -dependence in eq. (8). To be able to do
so, we restrict our analysis to eq. (9), assuming ε1,2 > 1, which avoids the problem of having to
estimate the value of N , or imposing an ad hoc choice of its order of magnitude. Using ε1 = ε2
in eq. (9), we recover the same functional form of the equilibrium distribution as in Kirman’s
article [18]. The distribution (9) exhibits a unique mode if both parameters take a value larger
than 1 (see panel a in Figure 1), while it shows bi-modality for the case ε1,2 < 1 (see panel
b). Furthermore, the distribution shows a monotonic behavior if one parameter is larger than 1
and the other smaller than 1 (see panels c and d for an example of a decreasing and increasing
monotonic distribution, respectively). The four equilibrium distributions with different choices
of ε1 and ε2 , shown in Figure 1, illustrate this great flexibility of the asymmetric model.4
[Insert Figure 1 approximately here]
An alternative way to describe our process (1) is via the Langevin equation:
p
zt+∆t = zt + ∆t[a1 − (a1 + a2 )zt ] + 2b ∆t (1 − zt )zt λt
(11)
where λt is a iid normally distributed random variable. In appendix A4 we show, that the
transition probability
ω(z 0 , t + ∆t|z, t) = N ω̄(n0 , t + ∆t|n, t)
(12)
can be written as a Gaussian with mean z +∆t A(z) and variance ∆t D(z) provided terms of the
order (ai ∆t)2 and (b∆t)2 can be neglected. Eq. (11) is another way to formulate this statement.
Usually (see Van Kampen [29] for the relationship between the Fokker-Planck and the Langevin
equation) an argument of this type is used to prove that (5) is a consequence of (11) in the
limit of ∆t → 0. In our case both (11) and (5) can be obtained from the Master equation for
large N and we do not have to worry about the validity of the Langevin equation at finite ∆t
(called Euler-Maruyama discretization in mathematical literature) as an approximation to the
Fokker-Planck equation in continuous time. Note that the restriction on the size of the time step
∆t in the Langevin equation is much less severe then condition (2) when using the transition
probabilities (1).
It is interesting to note that De Jong et al. [9] and Stegenborg, Larsen and Sørensen [28]
have employed the continuous time version of eq. (11) as a stochastic model of floating exchange
rates in a target zone. The model is estimated for several European FX rates using GMM and a
new estimation technique based on the eigenfunctions of the underlying Fokker-Planck equation
[16]. In contrast, our analysis derives eq. (11) as a consequence of an interacting-agent approach,
rather than as a starting point of a stochastic model for the description of the data.
3
The artificial financial market
3.1
Market implementation
We now implement this two-state dynamics as the main ingredient in a financial market model
with interacting heterogenous agents. Our market participants are divided into two groups:
4
Note that the symmetric model, characterized by a single parameter ε, exhibits a uni-modal distribution for
ε > 1, a bi-modal distribution ε < 1 and a uniform distribution for ε = 1. However, the monotonic case can not
be observed under the symmetric setting.
6
NF fundamentalists, who buy (sell) if the actual market price p is below (above) the fundamental value pF .
NC noise traders, who are subject to “irrational” fads and moods, following the seminal contribution by De Long et al. [10].
The arbitrary states 1 and 2 of the previous section are now identified as noise trader and fundamentalist strategies, respectively. This implies that a1 and a2 are the propensities for an agent
to switch autonomously to the other strategy, which might include a bias towards one of both
alternatives. For example, a1 > a2 would mean that on average the propensity of autonomous
conversion of a former fundamentalist to noise trader behavior is higher than the probability
for a switch in the opposite direction. The propensity to change behavior under the influence
of herding expressed by b must be the same in both directions, otherwise one would obtain a
trivial distribution for z.
In order to arrive at a formalization of the price dynamics, we have to formulate behavioral
rules for demand and supply of our two groups. Fundamentalists’ excess demand is given by:
EDF = NF ln
pF
.
p
(13)
Somewhat deviating from earlier contributions, we assume a dependence on the log difference
between the fundamental value and the current market price instead of absolute differences. This
appears intuitively plausible as fundamentalists’ trading should be based on relative rather than
absolute under- and overvaluation. Moreover, it facilitates derivation of a simple form of the process governing returns. Since the observed average daily changes are small (∼1%) we expect that
results would not be drastically different when using absolute differences instead of relative ones.
Excess demand of the noise traders is formalized as:
EDC = −r0 NC ξ
(14)
where ξ represents the average noise trader’s ‘mood’ (positive or negative). The constant r0
is a scale factor for their impact on the price formation, and the expression is multiplied by
−1 for notational convenience in the following derivations. Within a Walrasian scenario, the
equilibrium price is derived by setting EDF + EDC to zero. This leads to to the following
equation for the equilibrium price:
ln
NC
z
p
= r0
ξ = r0
ξ
pF
NF
1−z
(15)
where z and 1 − z are the fractions of noise traders and fundamentalists within the population,
respectively. Note that our choice of a random development of the noise traders’ mood ξt (and
derived from it, stochastic excess demand of this group) is inspired by the modelling of noise
traders in the seminal paper by de Long et al. [10]. Our noise traders, therefore, are characterized by random misperceptions rather than some heuristic trading rules such as trend following
behavior used, for instance, in Kirman’s adaptation of the ant model to foreign exchange trading
[17, 19].
7
We define the continuously compounded return over a time interval ∆t by
r(t) = ln
pt+∆t
.
pt
(16)
Using (15) we can write it as
r(t) = r0
z(t + ∆t)
z(t)
ξ(t + ∆t) −
ξ(t) .
1 − z(t + ∆t)
1 − z(t)
(17)
Empirically r is correlated only for time scales much smaller than a trading day. This property
can be build in by assuming that ξ changes much faster than z. Then the return (17) can
be approximated by considering only changes in ξ while keeping z constant. Defining η(t) ≡
ξ(t + ∆t) − ξ(t) this approximation yields:
r(t) = r0
z(t)
η(t).
1 − z(t)
(18)
The level of accuracy of this approximation depends on the choice of the underlying parameters, namely b and ε1,2 . For given values of ε1,2 , the accuracy between eqs. (17) and (18) is a
decreasing function of b, which, in fact, governs the diffusion part of the size of the increments
∆z (see eq. 11).
If we assume the random variable ηt to be iid on the time scale ∆t which we might think of
as daily data, daily returns in our model will be uncorrelated. Therefore, we model the development of the average noise traders’ mood ξ as a random walk with increments η. This choice
of the type of stochasticity governing the noise traders’ behavior is different from the related
literature, in particular the paper of De Long et al. [10], where the misperseption of the noise
traders itself is modeled as an iid random variable. The random walk assumption, however,
avoids the abrupt variations of the market price, which would occur if we would incorporate the
De Long formalization.
The behavior of r depends both on η(t) and the dynamics of z which is governed by the
herding mechanism. In the following we will discuss various choices for the distribution of η(t).
If sufficiently high moments of η exist, the power law property of the the distribution of z/(1−z)
derived in the next paragraph will also carry over to the distribution of returns.
[Insert Figure 2 approximately here]
In Figure (2) we show a typical simulated pattern of the price obtained from (15) using the
Langevin equation (11) for the change of z and a uniform distribution for ∆ξ = η. In the
lower parts we compare the returns calculated from (17) and from the approximation (18).
They behave qualitatively similar which can be taken as a justification of the approximation
(18)5 . One can observe large deviations from the constant fundamental price alternating with
small oscillations around it. This feature is related to the volatility clustering phenomenon
observed in the time series of returns. Large fluctuations of r happen when a large fraction of
traders adopts the noise trader behavior. On the contrary, in the presence of a large fraction of
5
Given a small value of b, the strict similarity of the two time series of returns in Figure (2) is very robust
with respect to the choice of ε1,2 .
8
fundamentalists only small variations occur. This intermittent dynamics apparently gives rise
to time-series properties capturing some of the stylized facts of empirical data The underlying
behavioral mechanisms are similar to those of the more involved models of Kirman [19], Lux
and Marchesi [25] and Wagner [32] from which our approach draws inspiration.
3.2
Analytical solutions for the unconditional distribution
The simple structure of eq. (18) permits to derive analytically the unconditional distribution of
returns6 . As a first step we write the return as
r = σ(t) · η(t)
with
σ(t) = r0
(19)
z(t)
.
1 − z(t)
(20)
The equilibrium probability distribution p(σ) can be computed from the general formula
p(σ) =
dz
· ω0 (z).
dσ
(21)
Using the relationship z = σ/(r0 + σ) derived from (20) and
dz
r0
=
dσ
(r0 + σ)2
we can express ω0 in equation (9) as function of σ and obtain the equilibrium distribution of σ
1
1
p(σ) =
r0 B(ε1 , ε2 )
σ
r0
ε1 −1 r0
σ + r0
ε1 +ε2
(22)
which exhibits a power law decay for σ r0
ε2 +1
1
p(σ) ∼
.
σ
(23)
As is obvious from eq. (23), the exponent of the tail is related to the parameters characterizing
the behavior of the fundamentalists, namely the ratio between the tendency a2 of autonomous
switches from fundamentalist to noise trader behavior, and the herding parameter b. The scaling
behavior of the tail also carries over to the distribution of r as shown in appendix A3. It is independent of the distribution of η provided the moment E[|η|ε2 ] exists7 . This remarkable property
of the equilibrium distribution of r is compatible with the empirical finding of power-law tails
described in the introduction.
Equation (19) also involves the noise term η. In order to obtain solutions in closed form for
the distribution of returns, we assume that η follows either:
6
From the Fokker-Planck equation (5), we can also derive further dynamical properties of returns, such as the
autocorrelation of the absolute values. The full analysis of the dynamics is left for future research.
7
In general, the tail behavior of a product of two independent random variables is governed by the component
with the smaller tail index (fatter tail).
9
• a bimodal distribution with two values ±1, occurring with equal probability (spin-noise
model ), or
• a uniform distribution over the interval [−1, +1] (uniform-noise model )
Since η is not directly observable, we could have chosen any other functional form, for instance a
Normal distribution or a Student t-distribution. Since the distribution of the noise is symmetric
around zero, we can confine our analysis to the distribution of absolute returns as a measure of
the volatility v:
v = |r| = σ|η|.
(24)
In the case of the spin-noise-model8 , the distribution of the volatility v is given by eq. (22), since
the absolute value of returns coincides with σ. The derivation of the distribution of absolute
returns in the case of uniformly distributed noise is more involved (see appendix A3), but can
also be obtained in closed form:
1 ε2
v
pu (v) =
1−β
; ε1 − 1, ε2 + 1
(25)
r0 ε 1 − 1
v + r0
where β is the incomplete beta function9 .
[Insert Figure 3 approximately here]
In Figure 3 we compare the probability density functions for the spin-noise-model and the
uniform-noise-model. To eliminate the influence of the scale parameter r0 we show the density
function for the normalized volatility vn = v/E[v]. The value of ε1 has been chosen such that
the second moment is the same in both cases. The behavior for v > E[v] is dominated by the
asymptotic eq. (23) and therefore very similar, as we expect from the discussion of eq.(22).
Both distributions mainly differ for small v, where the uniform noise model avoids convergence
to zero resulting from (22). Any continuous distribution of the noise generates a flat region or
‘plateau’ for small v in agreement with the data. This property will turn out to be the reason
for a better descriptive power of the uniform-noise model as compared to the spin noise model.
4
Estimation of the parameters of the models
To apply the model to empirical data we have to estimate the scale parameter r0 , the ratios
ε1,2 and the scale of the two constants a1 + a2 . The main difficulty is the occurrence of two
sources of randomness in the dynamics of returns in equation (24), the stochastic change of z
and the multiplicative noise η, whereas only one measurement (absolute value of the return) is
available at each time step. Therefore we divide the estimation into two steps. For a stationary Markov process with exponentially decaying correlations we can use the observed sample
of returns to estimate the equilibrium parameters r0 and ε1,2 by maximum likelihood fit to the
analytically known unconditional distributions. We should emphasize, however, that we use
only an approximation of the “true” likelihood; we pretend, in fact, that the realizations of the
Markovian process (18) are independent and identically distributed, according to the unconditional distributions given by equations (22) and (25), in the case of the spin noise or uniform
8
9
In the following, subscripts s and u indicate the spin-noise and uniform-noise
models, respectively.
Rx
The incomplete beta function is defined as β(x; a, b) = (1/B(a, b)) 0 ta−1 (1 − t)b−1 dt.
10
noise respectively. The advantage of using this approximation is the simplicity of its implementation and the reduced computational burden. On the other hand, only parameters from
the unconditional distribution can be estimated. The method gives asymptotically consistent
estimates as the sample size T → ∞, if the sampling frequency ∆ = ∆T → 0 and the increase of
T is faster than the decrease of ∆T , i.e. T ∆T → ∞. The estimates are asymptotically normal
under the additional condition T ∆2T → 0. (cf. Genon-Catalot et al. [11]). For an application
of this approximation to stochastic volatility models see [11]. Once intensities ε1,2 have been
estimated, the scale of a1 and a2 can be adjusted heuristically from the correlation of v(t). Since
the Langevin equation (11) has a drift term which is linear in z, the correlations of any function
of z have asymptotically an exponential decay with a characteristic time τ = (a1 + a2 )−1 . Since
η(t) in equation (24) is uncorrelated, this exponential decay carries over to v, with the same rate.
The second step of the estimation, therefore, consists in adjusting τ to the observed decay of the
correlation of v at large time lags. This method is not the most efficient, since the correlations
are not solely governed by the decay constant τ , but also depend on the herding parameter b
via the intensities ε1,2 .
Out of the three parameters that enter in the unconditional distribution, namely ε1 , ε2 and
r0 , the later can be expressed in terms of the other two parameters by imposing the following
normalization on the empirical data10 :
E[v] = 1.
(26)
Due to the relation E[z/(1 − z)] = ε1 /(ε2 − 1) equation (26) implies the following values of r0
which define the scale of the fluctuations:
ε2 − 1
1 spin noise
r0 =
(27)
·
2 uniform noise
ε1
For our implementation of the the estimation we use the following data sets: The daily returns of
the Gold price (5034 entries) over the period 1974-1998 (abbreviated by Gold), the daily quoted
stock prices of two large German companies, Deutsche Bank (DB, 6771) and Siemens (6674), in
the period 1974-2001 and the daily variations of the index of the German stock exchange (DAX,
9761) from 1959 to 1998.
4.1
Estimation of the spin-noise model
The resulting values of the parameters ε1 and ε2 in the case of the spin-noise-model are given
in Table 1.
10
The normalization is based on the implicit assumption of existence of the population mean. Therefore eq.
(27) holds under the condition ε2 > 1, which guarantees the existence of the mean of the process (20) (note that
ε2 governs the asymptotic behavior of the tail and, consequently, the existence of the moments).
11
Data Set
ε̂1
ε̂2
− ln Lε1 ,ε2
ε̂
− ln Lε
p-value
Gold
DB
Siemens
DAX
1.23 ± 0.04
1.53 ± 0.05
1.69 ± 0.06
1.37 ± 0.02
4.2 ± 0.4
5.1 ± 0.4
4.1 ± 0.1
14.0 ± 2.0
4926.6
6602.9
6457.4
9569.6
2.04 ± 0.02
2.44 ± 0.03
2.49 ± 0.03
2.50 ± 0.02
5081.9
6746.8
6540.0
10168.0
0.00
0.00
0.00
0.00
Table 1: Estimated parameters for the spin-noise-model ε1,2 (asymmetric model)
with the corresponding Log-Likelihood values at the maximum. The parameter ε
refers to the estimation with the constraint ε1 = ε2 = ε (symmetric model) with the
pertinent Log-Likelihood. The p-values of the Likelihood Ratio test are given in the
last column.
In Figure 4 we show the comparison of the theoretical distribution with the empirical one for
Gold and the DAX. For all the considered time series, except for DAX, we obtain a value of ε̂2
close to the empirical power law (see Table 2). The large entry in the case of DAX is due to the
extended plateau at v = 0. Since for ε1 > 1 the theoretical pdf has a zero at v = 0, the choice
of a large ε2 is the best compromise the estimation can make. A value ε1 < 1 would be even
worse since it leads to a spike at v = 0.
[Insert Figure 4 approximately here]
4.2
Estimation of the uniform-noise model
We also apply the maximum likelihood method to the same data sets to estimate the uniformnoise model. The resulting parameters are listed in Table 2.
ε̂1
ε̂2
− ln Lε1 ,ε2
ε̂
− ln Lε
p-value
α̂H
Gold
3.2 ± 0.4
3.9 ± 0.4
4942.7
3.6 ± 0.1
4943.2
0.34
DB
6.0 ± 1.0
4.4 ± 0.4
6683.0
4.9 ± 0.1
6683.7
0.25
Siemens
7.0 ± 2.0
3.9 ± 0.4
6573.9
4.7 ± 0.1
6576.8
0.02
16 ± 5
4.9 ± 0.3
9534.3
6.7 ± 0.2
9542.6
0.00
2.9
(2.4, 3.4)
3.4
(2.9, 4.0)
3.7
(3.2, 4.3)
3.1
(2.9, 3.6)
Data Set
DAX
Table 2: Estimated parameters using the uniform-noise model. Note somewhat different
results for the LR test. The last column shows the Hill estimates [14] of the index of the
tail for 2.5% tail size, with their 95% asymptotic confidence interval.
Figure 5 shows the fit for the Gold and the DAX. For the latter, there is a substantial improvement using the uniform-noise model over the spin-noise model, since the plateau is now correctly
accounted for. On the contrary, it seems that the goodness of fit is slightly worse for the other
cases, as we see from the lower values of the likelihood when compared to the pertinent values of
the spin-noise model. In appendix A5 we show that this somewhat disturbing conclusion, namely
12
the superior descriptive power of the spin-noise model over the uniform noise model, might be
just a fictitious effect of the discreteness of prices, which creates an artificial concentration of
small returns at certain threshold levels. Taking into account this particular feature of the data,
we, in fact, observe that the uniform-noise model performs better than the spin-noise framework.
The large values of ε̂1 , especially for DAX, are not unreasonable. They imply that fundamentalists have a relatively high propensity to autonomously switch to a noise trader behavior,
while the pertinent herding effect is comparatively smaller.
[Insert Figure 5 approximately here]
Recalling that the parameter ε2 should coincide with the so-called tail index of the unconditional
distribution, we compare the pertinent results to those obtained with the standard conditional
ML estimator of the tail index. As can be seen from Table 2, in all cases, the estimated values
of ε̂2 are somewhat below the 95% interval for the semi-parametric Hill estimator for the tail
index.11 The slight tendency of the parametric estimation towards somewhat lower values, when
compared with the semi-parametric estimator, may be explained by the influence of the center
of the distribution on the estimated values of ε2 . Nevertheless, we observe a relatively small
interval of variability of ε̂2 in harmony with the remarkable homogeneity of the tail index for
empirical data.
5
Discussion of the results
Our asymmetric herding model does account for the most pervasive stylized facts. The property
of fat tails of the distribution of returns is an essential consequence of our framework. This
is also true for volatility clustering, as shown in Figure 6. The clusters arise from equation
(18) with values z of the noise traders’ concentration getting close to 1. Due to the vanishing
diffusion term this situation may persist for some time. Figure 6 shows that the structure of
the autocorrelations for simulations of the model is, in fact, similar to that of empirical data in
many respects: (i) raw returns appear uncorrelated while (ii) squared and absolute returns have
highly significant autocorrelations which apparently fall off only slowly, (iii) autocorrelations of
absolute returns are larger throughout than those of squared returns. Absence of correlations
in raw returns has been build in by the assumption that the noise variable in equation (18) is
iid distributed. A short-coming of our model is the Markov property of herding models of this
type which implies that autocorrelations decay exponentially for large time lags. This precludes
long-memory strictu sensu (i.e., hyperbolic decline of the autocorrelations of some powers of
returns). However, an exponential decay with a small decay constant is hard to distinguish from
“true” long term-dependence (see Alfarano, Lux and Wagner for analytical details [2]).
[Insert Figure 6 approximately here]
Another important property of our model is that the parameters of the distribution can be related to the behavior of the agents. For example, ε1 > ε2 indicates that, on average, the market
is dominated by a noise trader attitude (i.e. E[z] > 0.5). However, irrespective of whether
11
One might remark, however, the well known fact that the asymptotic distribution underestimates the finitesample variability of estimates for processes with volatility clustering (cf [15]).
13
E[z] is above or below 0.5, the herding component leads to temporal emergence of a majority
of noise traders once in a while triggering large fluctuations which are, then, responsible for the
volatility clustering observed in Fig. 2. The asymmetry in the transition probabilities, therefore,
adds a considerable amount of flexibility to the original framework of the ants model in which
E[z] = 0.5 is imposed by assumption. A glance at Tables 1 and 2 shows that the spin-noise
model always indicates dominance of fundamentalists practices, while the uniform-noise model
only finds fundamentalists dominance in the case of Gold.
Note also that although ε2 exclusively governs the asymptotic tail behavior, the transition to
the asymptotic power law regime at large v is also affected by ε1 . For ε1 = 2, one finds an exact
Pareto law for pu (v) (see appendix A.3). With increasing ε1 , the extented tail region to which
the asymptotic power law applies decreases exponentially with ε1 . This implies the prediction
that markets with a strong herding component in the transition from fundamentalists to noise
traders have a pronounced power law behavior.
It seems interesting to compare our results to the estimates of the parameters of the Kirman model by Gilli and Winker [12] who found, in our notation, ε1 = ε2 ∼ 0.3. Despite the
differences in both the design of the herding mechanism and the market model, we might be
able to shed some light on the discrepancy in estimated parameters. In [12], a linear relation
between return (or price) and the fraction z of the noise traders is used, whereas we employ
the non-linear relation (18). However, to generate volatility clusters in a linear model, z has
to stay near the extreme values 0 or 1 most of the time, with a small probability to switch
between these modes (a detailed explanation of the underlying mechanism can be found in [1]).
In the symmetric herding model analyzed by Gilli and Winker, this can only be achieved with
values ε = ε1 ε2 < 1. In contrast, the nonlinearity and asymmetry of our model allows to obtain
volatility clustering without a bimodal distribution.
Finally, comparison of our asymmetric setting with the symmetric counterpart used in previous literature [18, 12], rises the question whether the data fully exploit this enhanced flexibility.
In order to shed some light on this issue, we have reported the outcome of likelihood ratio tests
in Tables 1 and 2, where the unconstrained model is described by the transition probabilities
(1) and the alternative is characterized by the constraint ε1 = ε2 = ε. As can be seen, the
outcome of the test depends to some extent on the type of multiplicative noise considered. In
the case of the spin-noise model the symmetric setting is strongly rejected by all data-sets. On
the contrary, the uniform-noise version shows less homogeneous behavior with only two cases of
rejection of ε1 = ε2 at 5% level of significance. Of course, the inability of rejection of ε1 = ε2 oc1
curs for cases in which we only find a slightly asymmetric distribution (e.g. E[z] = ε1ε+ε
= 0.46
2
in the case of gold) which is hard to distinguish from the symmetric case with a uni-modal
distribution (imposed by our assumption ε1,2 > 1). It would be interesting, therefore, to allow
for the bi-modal case (ε1,2 < 1) as an alternative which has a somewhat different mechanism of
generating volatility clustering (cf. Alfarano and Lux [1]). To include this scenario, we would,
however, have to take into account the N dependency of the limiting distribution of z (cf. eq.
(8)), which would translate into N dependency of v as well. Therefore, either an ad hoc choice
of the number of agents or some additional method of estimating N would be required. We
leave this topic for future research.
14
6
Conclusions
We have presented an extremely simple model of an agent-based artificial financial market, which
nevertheless is able to generate the key stylized facts of financial time series. Due to its very
simple structure, we were able to obtain a full analytical solution for the probability distribution
of returns, which allows a better understanding of the underlying dynamics. Consequently, the
parameters of the model can be estimated directly via maximum likelihood, giving more insights
into the potential behavioral origins of the statistical regularities of the data. One of the main
results of the paper is that we can directly connect some crucial micro variables of the market
(such as the traders’ tendency to switch between strategies) to a macroscopic observable (the tail
index). The estimations are quite encouraging, especially in the quality of fit of the unconditional
distribution. It has also been shown that asymmetry of the transitions between the two groups,
fundamentalists and noise traders, is not always rejected by the data in favor of a symmetric
framework. In conclusion, it appear worthwhile to include this further element of flexibility in
agent-based models of interaction in financial markets.
15
References
[1] S. Alfarano and T. Lux. A minimal noise traders model with realistic time series properties.
http://www.bwl.uni-kiel.de/ Ordnung+Wettbewerbspolitik/ewp/ewp.html, 2003. Working
paper, University of Kiel.
[2] S. Alfarano, T. Lux, and F. Wagner. Apparent long memory in agent-based models: an
explanation, 2003. Work in progress, University of Kiel.
[3] M. Aoki. New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics,
Multiple Equilibria, and Externalities as Field Effects. University Press, Cambridge, 1996.
[4] A. Beja and M. B. Goldman. On the dynamic behavior of prices in disequilibrium. Journal
of Finance, 35:235–248, 1980.
[5] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political
Economy, 81:637–653, 1973.
[6] W. A. Brock and C. H. Hommes. A rational route to randomness. Econometrica, 65:1059–
1095, 1997.
[7] R. Cont and J. P. Bouchaud. Herd behaviour and aggregate fluctuations in financial markets. Macroeconomic Dynamics, 4:170–196, 2000.
[8] D. Costantini and U. Garibaldi. A probabilistic foundation of elementary particle statistics.
Stud. Hist. Phil. Phys. 28, 28:483–506, 1997.
[9] F. De Jong, F. C. Drost, and B. J. M. Werker. A jump-diffusion model for exchange rates
in a target zone. Statistica Neerlandica, 55:270–300, 2001.
[10] B. J. De Long, A. Shleifer, L. H. Summers, and R. J. Waldmann. Noise trade risk in
financial markets. Journal of Political Economy, 98:703738, 1990.
[11] V. Genon-Catalot, T. Jeantheau, and C. Laredo. Parameter estimation for discretely observed stochastic volatility models. Bernoulli, 5:855–872, 1999.
[12] M. Gilli and P. Winker. A global optimization heuristic for estimating agent based models.
Computational Statistics and Data Analysis, 42:299–312, 2003.
[13] P. Gopikrishnan, M. Meyer, L.A.N. Amaral, and H.E. Stanley. Inverse cubic law for the
distribution of stock price variations. Eur. Phys. J. B, 3:139–140, 1998.
[14] B. M. Hill. A simple general approach to inference about the tail of a distribution. Annals
of Statistics, 3:1163–1173, 1975.
[15] P. Kearns and A.R. Pagan. Estimating the density tail index for financial time series.
Review of Economics and Statistics, 79:171–175, 1997.
[16] M. Kessler and M. Sø rensen. Estimating equations based on eigenfunctions for a discretely
observed diffusion process. Bernoulli, 5:299–314, 1999.
16
[17] A. Kirman. Epidemics of opinion and speculative bubbles in financial markets. In M. P.
Taylor, editor, Money and Financial Markets, pages 354–368. Blackwell, Cambridge, 1991.
[18] A. Kirman. Ants, rationality, and recruitment. Quarterly Journal of Economics, 108:137–
156, 1993.
[19] A. Kirman and G. Teyssiére. Microeconomic models for long memory in the volatility of
financial time series. Studies in Nonlinear Dynamics & Econometrics, 5:137–156, 2002.
[20] B. LeBaron. Agent based computational finance: Suggested readings and early research.
Journal of Economic Dynamics and Control, 24:679–702, 2000.
[21] I. N. Lobato and N. E. Savin. Real and spurious long-memory properties of stock market
data. Journal of Business and Economics Statistics, 16:261–283, 1998.
[22] T. Lux. The stable Paretian hypothesis and the frequency of large returns: An examination
of major German stocks. Applied Financial Economics, 6:463–475, 1996.
[23] T. Lux and M. Ausloos. Market fluctuations I: Scaling, multiscaling and their possible
origins. In A. Bunde, J. Kropp, and H. J. Schellnhuber, editors, Theories of Disaster
- Scaling Laws Governing Weather, Body, and Stock Market Dynamics, pages 373–409.
Springer, Berlin Heidelberg, 2002.
[24] T. Lux and M. Marchesi. Scaling and criticality in a stochastic multi-agent model of a
financial market. Nature, 397:498–500, 1999.
[25] T. Lux and M. Marchesi. Volatility clustering in financial markets: A micro-simulation
of interacting agents. International Journal of Theoretical and Applied Finance, 3:67–702,
2000.
[26] B. Mandelbrot. The variation of certain speculative prices. Journal of Business, 35:394–419,
1963.
[27] A. Pagan. The econometrics of financial markets. Journal of Empirical Finance, 3:15–102,
1996.
[28] K. Stegenborg Larsen and M. Sø rensen. Diffusion models for exchange rates in a target
zone. http://www.math.ku.dk/ michael/target.pdf, 2003.
[29] N. G. van Kampen. Stochastic processes in Physics and Chemistry. North Holland, Amsterdam, 1992. Revised edition.
[30] R. Vigfusson. Switching between chartists and fundamentalists: A Markov regime-switching
approach. International Journal of Finance and Economics, 2:291–305, 1997.
[31] C. G. de Vries. Stylized facts of nominal exchange rate returns. In F. van der Ploeg, editor,
The Handbook of International Macroeconomics, pages 348–389. Blackwell, Oxford, 1994.
[32] F. Wagner. Volatility cluster and herding. Physica A, 322:607–619, 2003.
[33] T. Weiland. On the unique numerical solution of maxwellian eigenvalue problems in three
dimensions. Particle Accelerators, 17:227242, 1985.
17
[34] F.H. Westerhoff and S. Reitz. Nonlinearities and cyclical behavior: The role of chartists
and fundamentalists. Studies in Nonlinear Dynamics and Econometrics, 7, 2003.
[35] M. Youssefmir and A. Huberman. Clustered volatility in multiagent dynamics. Journal of
Economic Behavior and Organization, 32:101–118, 1997.
[36] J. Zinn-Justin. Quantum Field Theory and Critical Phenomena. Clarendon Press, Oxford,
1989.
A
Appendix
A.1
Derivation of the Fokker-Planck equation
To derive the Fokker-Plank equation (5) we start by writing the transition probabilities from
eq. (1) for the switching process as:
π(n → n + 1) = π+ (n) = (N − n)(a1 + bn)
(28)
π(n → n − 1) = π− (n) = n[a2 + b(N − n)].
(29)
The transition probabilities define a Markovian stochastic process, that belongs to the class of
so-called “birth-death processes” [3] or “one-step processes” [29]. The so-called Master equation
for the flux of probability from eq. (3) reads12
∆ω̄n (t)
= ω̄n+1 π− (n + 1) + ω̄n−1 π+ (n − 1) − ω̄n π− (n) − ω̄n π+ (n).
∆t
Defining the current j¯n as the probability flow from n − 1 to n
j̄n = ω̄n−1 π+ (n − 1) − ω̄n π− (n)
(30)
(31)
the Master equation can be written as a discrete continuity equation:
∆ω̄n
+ j̄n+1 − j̄n = 0
∆t
(32)
P
One proves from (32) that n ω̄n = 1 holds for all t if it holds at t = 0, provided the current
vanishes at the boundaries (j̄0 = j̄N +1 = 0) for any t. This explains the term ’continuity’. For
N 1 we introduce the continuous variable z = n/N . Its probability density ω(z) is defined as
ω(z) = N ω̄n
with z =
n
N
(33)
R
The normalization factor N ensures that ω(z)dz = 1 holds up to terms of order N −2 . In
the following we will neglect those terms. We also replace (31) for the current by a continuous
function j(z):
1
j̄n = j(z −
).
(34)
2N
The offset 1/(2N ) in the argument of j(.) in eq. (34) is a commonly used technique in gauge
theories on a discrete lattice [36] or the discretization of Maxwell’s equations for electric phenomena in numerical simulations [33]. It expresses that j̄n connects the points n − 1 and n and
12
In the following we omit the obvious time argument t in ω̄n and j̄n .
18
therefore j(z) should be defined in the middle of (n − 1)/N and n/N as is done in equation (34).
We plug (34) into the difference j̄n+1 − j̄n :
j̄n+1 − j̄n = j(z +
1
1
) − j(z −
).
2N
2N
Using a third order Taylor expansion for j(z) even powers in 1/N all vanish, and we get
1
1
0
000
j (z) +
j (z) .
j̄n+1 − j̄n =
N
24N 2
(35)
The continuum version of the Master equation (32) is obtained by inserting the difference (35)
and the definition (33) into (32) neglecting terms of order 1/N 2
∆ω(z)
∂j(z)
+
= 0.
∆t
∂z
(36)
It is important to define the current (34) with the offset, otherwise we would have got terms
of order 1/N which are cancelled in a tedious calculation by corresponding corrections in j(z)
lateron.
Now we express j(z) in terms of ω(z). Rewriting equation (31) for j(z) using (34) yields:
j(z −
1
) = ω̄n−1 π+ (n − 1) − ω̄n π− (n)
2N
and using the definition (33) we obtain:
1
1
1
j(z −
)=
ω(z − ) · π+ (zN − 1) − ω(z) · π− (zN ) .
2N
N
N
In this relation only continuous functions appear and we can shift the variable z by 1/(2N ) to
obtain:
1
1
1
1
1
ω(z −
) · π+ (zN − ) − ω(z +
) · π− (zN + ) .
(37)
j(z) =
N
2N
2
2N
2
Up to terms of order 1/N 2 we can use the approximation ω(z ± 1/(2N )) = ω(z) ± 1/(2N )ω 0 (z)
and get
j(z) = ω(z)
π+ (zN − 12 ) − π− (zN + 21 )
π+ (zN − 21 ) + π− (zN + 12 )
− ω 0 (z)
.
N
2N 2
(38)
π± (n) from equation (31) can be written as functions of z and N . Introducing the function D(z)
D(z) = 2bz(1 − z) +
1
(a1 − z(a1 − a2 ))
N
(39)
the coefficient in the second term in eq. (38) can be written as
π+ (zN − 21 ) + π− (zN + 12 )
a1 + a2 − b
= D(z) +
.
N2
N2
(40)
A(z) = a1 − z(a1 + a2 )
(41)
With the definition
19
the first coefficient in (38) can be expressed in terms of A and D as:
π+ (zN − 12 ) − π− (zN + 21 )
1
= A(z) − D0 (z).
N
2
(42)
Inserting the expressions (42) and (40) without the 1/N 2 term into (38) we finally get
j(z) = A(z)ω(z) −
1 ∂
(D(z)ω(z)).
2 ∂z
(43)
Since ∆t is also of order 1/N 2 the difference in time can be replaced by a partial derivative
which, together with the current (43), leads to the Fokker-Planck equation (5). The structure of
the Fokker-Planck equation is correct up to terms of order 1/N 2 which involve higher derivatives
of ω(z). It is remarkable that no contribution of the herding mechanism to the drift term A(z)
appears and the 1/N correction shows up only in the diffusion term D(z).
A.2
Unconditional distribution of z
To compute the equilibrium distribution ω0 (z) in eq. (8), we use the standard formula:
Z z
K
2A(y)
ω0 (z, N ) =
exp
dy
D(z)
D(y)
(44)
(see for instance Van Kampen [29]), derived from the condition ∂j(z, t)/∂z = 0. The drift term
D can be written up to order 1/N 2 as a product
ε1 ε2 D(y) = 2b y +
1−y+
.
2N
2N
Writing A(y) as
h ε1 i
ε2 − ε2 y +
,
A(y) = b ε1 1 − y +
2N
2N
the integral in the argument of the exponential is elementary with the result
Z z
2A(y)
ε1 ε2 dy = ε1 ln z +
+ ε2 ln 1 − z +
.
D(y)
2N
2N
(45)
Inserting this integral into (44) we arrive at the following formula for ω0 (z, N ):
ε1 ε1 −1 ε2 ε2 −1
ω0 (z, N ) = K(ε1 , ε2 , N ) z +
· 1−z+
(46)
2N
2N
R
The normalization constant K follows from ω0 (z, N ) dz = 1. If we neglect the 1/N term in D
we can express K as the beta function:
Z 1
Γ(ε1 )Γ(ε2 )
B(ε1 , ε2 ) =
z ε1 −1 (1 − z)ε2 −1 =
(47)
Γ(ε1 + ε2 )
0
which leads to the distribution given in (9).
20
A.3
Unconditional distribution of uniform-noise model
The volatility v is a product of the noise variable |η| distributed according to ρ(η) and σ distributed according to p(σ) from equation (22). For the distribution of v under the condition of
a given value of σ we have
d|η| v
2 v
p(v|σ) = 2
ρ( ) = ρ( ).
(48)
dv
σ
σ σ
The unconditional distribution of v is obtained by integrating p(v|σ) over σ with the probability
distribution p(σ)
Z
pu (v) = 2
p(v|σ) p(σ) dσ
(49)
Z
1 v
ρ( ) p(σ) dσ.
(50)
= 2
σ σ
For a uniform distribution ρ(η) we get
Z
∞
pu (v) =
v
1
p(σ) dσ.
σ
Inserting p(σ) we find
1
pu (v) = 2
r0 B(ε1 , ε2 )
Z
v
∞
u
r0
ε1 −2 r0
u + r0
ε1 +ε2
du.
(51)
If we we make the substitution u = r0 t/(1 − t) the integral can be reduced to the incomplete
beta function β defined by:
Z x
1
β(x; a, b) =
ta−1 (1 − t)b−1 dt
(52)
B(a, b) 0
with B given in (47). Using the recursion formula bB(a, b) = (a − 1)B(a − 1, b + 1) this leads to
eq. (25):
1 ε2
v
pu (v) =
; ε1 − 1, ε2 + 1 .
1−β
(53)
r0 ε 1 − 1
v + r0
If either ε1 or ε2 is an integer the integral in (51) can be solved explicitly. In the case ε1 = 2 we
obtain a pure Pareto law
ε2 +1
ε2
r0
pu (v) =
.
(54)
r0 v + r0
The case ε1 = 3 leads to a distribution with the same asymptotic behavior but modified at small
and intermediate v:
ε2 +1
ε2
r0
(ε2 + 2)v + r0
pu (v) =
.
(55)
r0 v + r0
2(r0 + v)
To prove the power law behavior of pu (v) for any ρ(η) we rewrite equation (49) using the
symmetry ρ(η) = ρ(−η)
Z ∞
dη
v
pu (v)v ε2 +1 = 2
ρ(η) p( ) v ε2 +1 .
(56)
η
η
0
21
The integral is well defined for any v. If the limit v → ∞ exists we can interchange the limit
v → ∞ with the integration. Since for large v we have
v
1
p( )v ε2 +1 '
(r0 η)ε2 +1
η
r0 B(ε1 , ε2 )
(57)
we find the limiting behavior
lim pu (v)v
ε2 +1
v→∞
2
=
B(ε1 , ε2 )
Z
∞
dη (r0 η)ε2 ρ(η).
(58)
0
This limit exists if the ε2 -th moment of ρ(η) exists. If the moment does not exist, the asymptotic
behavior is dominated by ρ(η) for large |η|.
A.4
Proof of the Langevin equation
To prove equation (11) we begin with the following moments of the transition probability ω̄(n0 , t+
∆t|n, t):
X
k
Mk (n) =
n0 − n ρ̄(n0 , t + ∆t|n, t).
(59)
n0
Inserting the probabilities (28) and (29) we get
M0 (n) = 1
(60)
Mk (n) = ∆t
π+ (n) − π− (n) for k > 0
π+ (n) + π− (n) for k > 0
odd
even.
Introducing the continuum transition probability
ρ(z 0 |z) = N ρ̄(n0 , t + ∆t|n, t)
its moments are given by
Z
dz 0 (z 0 − z)k ρ(z 0 |z) = N −k Mk (zN ).
The characteristic function C(q) = E[exp(iq(z 0 − z))|z] can be computed from the moments
C(q) =
∞
X
(iq)k
k=0
k!
Mk (zN )
(61)
∞
X
2k+1
1
iq
= 1 + ∆t (π+ (zN ) − π− (zN ))
(2k + 1)! N
k=0
∞
X 1 iq 2k
+∆t (π+ (zN ) + π− (zN ))
.
(2k)! N
k=1
Using the identities
π+ (zN ) − π− (zN ) = N A(z)
and
22
π+ (zN ) + π− (zN ) = N 2 D(z)
(62)
(63)
and neglecting terms of order ∆t/N k with k ≥ 2 we find
q2
C(q) = 1 + ∆t iqA(z) − D(z) .
2
(64)
If we further assume ai ∆t 1 and b∆t 1 this characteristic function C corresponds to
a Gaussian for ω(z 0 |z). This implies that z 0 for given z is Gaussian distributed with mean
z + ∆t A(z) and variance ∆t D(z), which is the content of (11). Note that the above inequalities
are less restrictive than (2).
A.5
Discreteness of price records
In the estimation procedure, an important issue arises from the discrete nature of the records
of prices. Close inspection of our empirical time series shows a variability of the precision of
the entries that changes over time. The DAX, for instance, exhibits a decimal precision for
the first 7064 data points, which is increased to the second digit afterwards; the price of Gold
is recorded, for a long period, with a precision of a quarter of dollar, but changes to smaller
increments later on. The discreteness of prices is reflected in an artificial large variability of small
returns (|r| < E(|r|)), where certain values are more frequent than others, due to some sort of
threshold effect. Consequently, the empirical proxy for volatility |r| is poorly approximated by a
continuous variable in the region close to zero, while the key variable in our model is continuous
over its entire range (see eq. 24). In order to induce continuity, one could add a small white
Gaussian noise with mean zero to the time series of prices, to avoid the discreteness of small
returns. If one takes a standard deviation of the noise of 0.05, this should be sufficiently small to
just marginally affect the original price level, but large enough to avoid the discreteness in the
last decimal13 . To evaluate the impact of the noise in the estimation of ε1 , ε2 and the values of
the likelihood, we have performed 100 Monte Carlo simulations for different realizations of the
noise entries for all the time series, and for the case of both the uniform and spin noise models.
It turns out that the estimates are not very different across replications of this Monte Carlo
setting. The resulting standard deviation of the parameters ε1 and ε2 is, in fact, much smaller
than the associated error of a single estimate due to the Maximum Likelihood method. For
example, in the case of Gold and spin-noise model, the average value of ε1 over 100 simulation
is 4.7 and the standard deviation is 0.1, while the error associated to the ML estimate is 0.4.
(further details upon request). Therefore, estimates from a particular realization of the modified
data sets should be quite representative. Table 3 shows representative numbers from one such
particular realization.
A comparison of Tables 1 and 2 with Table 3 shows that when adding noise to the prices, the
uniform-noise model is preferred in all cases over the spin-noise model on the basis of its higher
likelihood value. Therefore, the superior performance of the simpler multiplicative spin-noise in
describing the raw data seems, in fact, to be due to the discreteness of the price records. The
additive noise, in fact, destroys the artificial abrupt variations of small returns, generating a flat
region around the origin, which, then, is responsible for the large entries of ε2 estimates in the
spin-noise model, cf. Table 3 (except for Gold case, which does not exhibit such an extended
13
In the case of Gold price the standard deviation is 0.1, since the minimum increment is 1/4.
23
Spin
Data Set
ε̂1
ε̂2
− ln Lε1 ,ε2
ε̂
− ln Lε
p-value
Gold
DB
Siemens
DAX
1.14 ± 0.03
1.20 ± 0.02
1.22 ± 0.03
1.34 ± 0.02
4.8 ± 0.4
13.0 ± 2.0
16.0 ± 3.0
15.0 ± 2.0
4944.9
6726.2
6621.6
9592.0
1.99 ± 0.02
2.24 ± 0.02
2.30 ± 0.03
2.46 ± 0.02
5181.2
7228.5
7117.0
10261.1
0.00
0.00
0.00
0.00
3.3 ± 0.4
7.2 ± 1.6
7.3 ± 1.5
15 ± 5
3.9 ± 0.4
4.7 ± 0.5
5.0 ± 0.5
4.9 ± 0.3
4943.2
6691.5
6590.9
9533.9
3.6 ± 0.1
5.4 ± 0.2
5.7 ± 0.2
6.7 ± 0.2
4943.5
6292.6
6591.8
9542.8
0.43
0.13
0.18
0.00
Gold
DB
Uniform
Siemens
DAX
Table 3: Estimated parameters for the spin-noise-model and the uniform-noise model with the
additive noise in the time series of prices.
plateau). A close inspection of the estimates shows that the uniform-noise model is more robust against additive noise in the prices than the spin-noise model, as its estimates are only
marginally affected by the modification.
Finally, this estimation exercise reveals that discreteness of the price levels, even for daily
data, can have a sizeable influence on statistical estimates. The effect is especially noticeable if
one works with parametric models capturing the entire distribution, since in such models small
returns constitute a large fraction of the sample size.
24
Figure 1: The four panels show different probability densities derived from eq. (9) for several
choices of the parameters ε1 and ε2 . In the panel (a) a uni-modal distribution, in (b) a bimodal distribution and in panels (c) and (d) two cases of asymmetric monotonic distributions
are shown.
25
Figure 2: The upper panel shows a simulation of the price derived from equation (15) using a
uniform distribution of η over [−0.1, +0.1]. In the middle panel the returns corresponding to
(17) are shown. The lower panel shows the returns obtained by using eq. (18). As parameters
we have chosen pf = 1, r0 = 0.1 and ∆t = 1. The herding parameters ε1 = 3, ε2 = 4 and
b = 0.003 are representative for the case of the price of Gold.
26
Figure 3: The figure compares the probability density functions of the spin-noise-model and the
uniform-noise-model as functions of the normalized volatility vn = v/E[v]. The parameters for
the spin-noise-model are ε1 = 1.8 and ε2 = 4, for the uniform-noise-model ε1 = 6 and ε2 = 4.
For this choice of ε1 both densities have the same mean and second moment.
27
(a) Gold
(b) DAX
Figure 4: Spin-noise model: In panel (a) the probability density of the time series of Gold
is shown as a function of the normalized volatility vn together with the distribution (22) with
estimated parameters given in Table 1. The inlet shows for both cases the cumulative distribution
P (|r| > x) in a log-log plot. Panel (b) shows the same comparisons for the time series of DAX.
Note the deviation of the theoretical curve in the outer part of the cumulative distribution and
the poor fit in the central part of the pdf in the case of the DAX. The graph also shows intervals
of ±1x standard deviation, which are computed assuming a normal distribution for the entries
in every bin of the histogram. The same procedure is applied also in Figure 5.
28
(a) Gold
(b) DAX
Figure 5: Uniform-noise model: In panel (a) the probability density of the time series of Gold
is shown as a function of the normalized volatility vn together with the distribution (25) with
estimated parameters given in Table 2. The inlet shows the cumulative distribution P (|r| > x)
in a log-log plot. Panel (b) shows the fit for the time series of DAX. In contrast to the spin-noise
model the uniform-noise model gives a good fit for all vn .
29
Figure 6: The upper panel shows simulated returns; autocorrelation function of raw, absolute
and squared returns are exhibited in the lower panel. Underlying parameters are: b = 0.0025,
ε1 = 16, ε2 = 4.9, r0 = 1 and pF = 1. The time series is derived from (18) using the uniformnoise model and z(t) simulated via the Langevin equation (11). ε1 and ε2 are representative for
the case of DAX.
30